Question

a recent survey found that 70% of all adults over 50 wear glasses for driving. in a random sample of 10 adults over 50, what is the probability that at least six wear glasses?
a. 0.700
b. 0.850
c. 0.006
d. 0.200

Explanation:

Step1: Recall binomial probability formula

The binomial probability formula is \(P(X = k)=C(n,k)\times p^{k}\times(1 - p)^{n - k}\), where \(n = 10\), \(p=0.7\), \(C(n,k)=\frac{n!}{k!(n - k)!}\). The probability of at least six is \(P(X\geq6)=P(X = 6)+P(X = 7)+P(X = 8)+P(X = 9)+P(X = 10)\)

Step2: Calculate \(P(X = 6)\)

\(C(10,6)=\frac{10!}{6!(10 - 6)!}=\frac{10\times9\times8\times7}{4\times3\times2\times1}=210\)
\(P(X = 6)=C(10,6)\times(0.7)^{6}\times(0.3)^{4}=210\times0.117649\times0.0081 = 0.2001\)

Step3: Calculate \(P(X = 7)\)

\(C(10,7)=\frac{10!}{7!(10 - 7)!}=\frac{10\times9\times8}{3\times2\times1}=120\)
\(P(X = 7)=C(10,7)\times(0.7)^{7}\times(0.3)^{3}=120\times0.0823543\times0.027 = 0.2668\)

Step4: Calculate \(P(X = 8)\)

\(C(10,8)=\frac{10!}{8!(10 - 8)!}=\frac{10\times9}{2\times1}=45\)
\(P(X = 8)=C(10,8)\times(0.7)^{8}\times(0.3)^{2}=45\times0.05764801\times0.09 = 0.2335\)

Step5: Calculate \(P(X = 9)\)

\(C(10,9)=\frac{10!}{9!(10 - 9)!}=10\)
\(P(X = 9)=C(10,9)\times(0.7)^{9}\times(0.3)^{1}=10\times0.040353607\times0.3 = 0.1211\)

Step6: Calculate \(P(X = 10)\)

\(C(10,10)=\frac{10!}{10!(10 - 10)!}=1\)
\(P(X = 10)=(0.7)^{10}=0.0282\)

Step7: Sum up the probabilities

\(P(X\geq6)=0.2001 + 0.2668+0.2335 + 0.1211+0.0282=0.85\)

Answer:

B. \(0.850\)